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An equilateral triangle and a square are inscribed in a circle as shown. ABCis isosceles. The triangle and square share a common vertex. What is the number of degrees in the measure of the angle indicated by the question mark?

https://latex.artofproblemsolving.com/5/0/1/501edee562d2ddbfd5f0d6209c480fb332a0dea7.png

Guest Dec 4, 2018

edited by
Guest
Dec 4, 2018

#1**+2 **

The triangle with the ? in it has a 90 degree angle , a ? angle and a 15 degree angle in it (the upper 90 angle of the square has 60 degrees taken away from it, leaving the remainder split between two small angles on either side of the 60)..

Does this help?

ElectricPavlov Dec 4, 2018

#3**+1 **

Why are you confused? EP said that the angle at the vertex that they(the equilateral triangle and the square share) is a right angle, or 90 degrees . Then the middle angle of the equilateral triangle is 60 degrees. Why? Because "equilateral" triangle has 3 "equal" sides and, therefore, 3 "equal" angles!!. Since the sum of the 3 angles of ANY triangle is 180 degrees, then 180/3 =60 degrees for each angle. Since the angle at the vertex is "splitting" a right angle, or 90 degrees which belongs to the square, then that leaves 2 smaller angles on each side of the 60-degree angle, or 90 - 60 =30/2 =15 degrees. That is the top angle of the triangle you are trying to solve.

Since that small triangle is a RIGHT triangle, then: 180 - 90 - 15 =75, which is the angle with a question mark?

Got it now?

Guest Dec 4, 2018